1. The Field of the Invention
The present invention relates to systems, methods and training aids for teaching the formulation of mathematical word problems, and more particularly relates to a method of sequentially constructing and displaying the mathematical components associated with a word problem.
2. The Prior State of the Art
The classic research of the famous mathematician George Polya (from Stanford University) characterizes the problem solving techniques used in current education literature. The strategies (or heuristics) identified by Polya include:
considering similar problems, PA1 introducing symbols or representations, PA1 drawing pictures and diagrams, PA1 work backwards from the desired results, PA1 reducing the gap between what is known and that which is not known, and PA1 breaking up complex problems into simpler ones. PA1 A `unit` is a single measurement or quantifiable entity, for example, ft, gallons, lbs, dollars ($), miles, etc. PA1 A `dimension` is a single unit or any combination (multiplied or divided) of units, for example, miles/hr, lbs/inch.sup.2, $/package, miles/gallon etc. (In the literature, units and dimensions are often used interchangeably.) PA1 A `primary component` is a single value or a single symbol representing a constant or a variable (unknown), for example, 15 miles/gallon, G gallons, etc. PA1 A `component` is a primary component or any combination (using mathematical operators) of primary components, for example, 15 G miles (which is the number of miles that can be traveled with G gallons. PA1 `Dimensional analysis` (i.e., `units analysis`) is the technique of verifying proper combining and relating of components through the cancellation of common units in the denominator and numerator when multiplying components and through the restriction of adding or comparing only those components with similar dimensions. PA1 A `schematic diagram` is a drawing that illustrates the relationship between a specific group of components. For example, the above components concerning gallons per mile could be illustrated on a map by placing hash marks every fifteen miles and numbering them in sequence. PA1 A `conversion component` is a known relationship between two or more units. For example: 5280 ft/mile, 16 oz/lb, etc. PA1 A `formula` is a known mathematical relationship between two or more entities. For example, the length C of the hypotenuse of a right triangle satisfies the formula C.sup.2 =A.sup.2 +B.sup.2 where A and B are the lengths of the other sides. PA1 The `tool box` is a collection of mathematical facts that the student has available to them to construct the components. Such facts include conversion components, formulas, axioms, theorems, rules, and hints on how certain words are translated into mathematical statements. PA1 A `functional relationship` is a mathematical expression that determines the relationship of one component to other components. Examples include equations, inequalities, probability distributions, etc. PA1 Identify a primary component in the word problem with its corresponding dimension and description. PA1 Identify a conversion component whose dimension contains a unit contained in the dimension of a previously defined component. PA1 Identify a (compound) component that can be constructed from previously defined components using mathematical operators. PA1 Identify a functional relationship between the components defined above using schematic diagrams and tools from the tool box.
These strategies are purposefully general to characterize various types of problems. The current state of the art is to apply these various strategies also to a subset of problems, called word problems. The traditional use of a word problem is to illustrate the abstract mathematical principles covered in the current material being studied, say, in the previous section or chapter. Students focus their attention on the abstract concepts and associate those concepts with the related words in the given problem. Very seldom does a student develop a structured approach to word problems in general, and unfortunately, most students develop a fear of word problems over the course of their education.
Various studies indicate successful problem solving depends not only on having various strategies at our disposal, but on the knowledge base of specific mathematical tools. Such tools are acquired gradually as the student develops a complex network of mathematical ideas and processes, in some sense, a tool box. The acquisition of these tools and the ability to formulate problems should be nurtured together. The importance of formulating word problems is emphasized by the fact that The National Council of Teachers of Mathematics has cited problem solving as one of their major goals over the last couple of decades.
Dimensional analysis is a technique used primarily in the physical sciences to verify proper construction of quantitative expressions. The technique is based on the cancellation of common units in the denominator and numerator when multiplying components and through the restriction of adding and comparing only those components with similar dimensions. For example the gravitational constant g has the dimension of meters/sec.sup.2. So to find the velocity of a falling object after 10 seconds, one can express it as EQU (10 sec)(g meters/sec.sup.2)=10 g meters/sec
Notice, in particular, the sec in the numerator of the dimension of 10 canceled with one of the sec units in the denominator of dimension of g. The resulting dimension indeed reflects the nature of velocity.
Some authors have used the concept of dimensional analysis to help students understand "ratio problems". In U.S. Pat. No. 5,441,278 (Apparatus and Method of Playing an Educational Card Game), the inventor accurately stated that very few schools teach this technique. Existing products currently on the market (such as Mathcad from Mathsoft) emphasize the solutions, graphs, and computational aspects of mathematics. They use dimensional analysis only to verify mathematical expressions. Dimensional analysis has not been taught widely, because it is not been applied to the extent that this invention introduces.
As typified by the success of the World Wide Web (Web), hypertext has played a significant role in organizing information and creating an educational environment. In fact, the primary protocol on the Web is HTML, which stands for HyperText Markup Language. Due to the lack of a structured approach to formulating word problems, the use of hypertext techniques has not been applied systematically to the formulation of word problems.
In summary, there does not exist a comprehensive method used as the fundamental basis for identifying and constructing the various components of word problems. Therefore current educational institutions do not teach a single systematic framework to formulate such problems, and most students are frustrated with their ability to understand and formulate word problems. Dimensional analysis is used only in a limited way and a centralized accessible depository of mathematical facts for the student to effectively draw from when formulating word problems has not materialized. Finally, without a structured methodology, there is no training tools for the systematic formulation of word problems based on hypertext technologies.